Simulation and estimation problems associated with the 3-parameter gamma density

Abstract

A new algorithm is introduced for the solution of the three parameter model, in which s , a , p refer to the maximum likelihood estimators of the threshold, scale, and shape parameters respectively. This basically pivots on an equation for the threshold estimator. It seems possible that in finite samples moments of p and s do not exist and indeed there is some doubt about the definition of asymptotic moments. However the statistic ln(p ) (p ) is less chaotic, and has low order moments not far removed from the normal and in some cases approximately X2 tyPe- Fr the maximum likelihood estimators we only consider cases for which p 2. When p 2, modified maximum likelihood estimators introduced by Cohen and Whitten are discussed. Four moments for each of the threeestimators are simulated, and from these non-extreme percentage points assessed.

Keywords:

• Existence of maximum likelihood estimator
• four moment distributional approximants
• location problems
• singularities
• stability of simulations
• transformations